Multi-winged Lorenz attractors due to bifurcations of a periodic orbit with multipliers $(-1,i,-i)$

Abstract

We show that bifurcations of periodic orbits with multipliers $(-1,i,-i)$ can lead to the birth of pseudohyperbolic (i.e., robustly chaotic) Lorenz-like attractors of three different types: one is a discrete analogue of the classical Lorenz attractor, and the other two are new. We call them two- and four-winged ``Sim\'o angels''. These three attractors exist in an orientation-reversing, three-dimensional, quadratic H\'enon map. Our analysis is based on a numerical study of a normal form for this bifurcation, a three-dimensional system of differential equations with a Z4-symmetry. We investigate bifurcations in the normal form and describe those responsible for the emergence of the Lorenz attractor and the continuous-time version of the Simo angels. Both for the normal form and the 3D H\'enon map, we have found open regions in the parameter space where the attractors are pseudohyperbolic, implying that for every parameter value from these regions every orbit in the attractor has positive top Lyapunov exponent.